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SageMath
E = EllipticCurve("bb1")
E.isogeny_class()
Elliptic curves in class 38025bb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 38025.cm6 | 38025bb1 | \([1, -1, 0, -4183542, -3292403009]\) | \(147281603041/5265\) | \(289471654713515625\) | \([2]\) | \(774144\) | \(2.4396\) | \(\Gamma_0(N)\)-optimal |
| 38025.cm5 | 38025bb2 | \([1, -1, 0, -4373667, -2976605384]\) | \(168288035761/27720225\) | \(1524068262066659765625\) | \([2, 2]\) | \(1548288\) | \(2.7862\) | |
| 38025.cm7 | 38025bb3 | \([1, -1, 0, 7984458, -16755914759]\) | \(1023887723039/2798036865\) | \(-153837105652605457265625\) | \([2]\) | \(3096576\) | \(3.1328\) | |
| 38025.cm4 | 38025bb4 | \([1, -1, 0, -19773792, 31011470491]\) | \(15551989015681/1445900625\) | \(79496153175699228515625\) | \([2, 2]\) | \(3096576\) | \(3.1328\) | |
| 38025.cm8 | 38025bb5 | \([1, -1, 0, 23004333, 146811854866]\) | \(24487529386319/183539412225\) | \(-10091065026005091847265625\) | \([2]\) | \(6193152\) | \(3.4793\) | |
| 38025.cm2 | 38025bb6 | \([1, -1, 0, -308953917, 2090263140616]\) | \(59319456301170001/594140625\) | \(32666072146490478515625\) | \([2, 2]\) | \(6193152\) | \(3.4793\) | |
| 38025.cm3 | 38025bb7 | \([1, -1, 0, -301539042, 2195354163991]\) | \(-55150149867714721/5950927734375\) | \(-327184216210842132568359375\) | \([2]\) | \(12386304\) | \(3.8259\) | |
| 38025.cm1 | 38025bb8 | \([1, -1, 0, -4943250792, 133773808843741]\) | \(242970740812818720001/24375\) | \(1340146549599609375\) | \([2]\) | \(12386304\) | \(3.8259\) |
Rank
sage: E.rank()
The elliptic curves in class 38025bb have rank \(0\).
Complex multiplication
The elliptic curves in class 38025bb do not have complex multiplication.Modular form 38025.2.a.bb
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 8 & 8 \\ 8 & 4 & 8 & 2 & 4 & 1 & 2 & 2 \\ 16 & 8 & 16 & 4 & 8 & 2 & 1 & 4 \\ 16 & 8 & 16 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
